KK w/ lots of flop action

No real reads yet. This one confused me. Thought I might be up against a set or maybe even a straight. on the flop. I got scared and went into check call mode. How should I have played this.

PokerStars 1/2 Hold'em (9 handed) converter

Preflop: Hero is SB with K[img]/images/graemlins/spade.gif[/img], K[img]/images/graemlins/heart.gif[/img].
<font color="#666666">5 folds</font>, CO calls, <font color="#666666">1 fold</font>, <font color="#CC3333">Hero raises</font>, BB calls, CO calls.

Flop: (6 SB) 4[img]/images/graemlins/spade.gif[/img], 3[img]/images/graemlins/heart.gif[/img], 6[img]/images/graemlins/spade.gif[/img] <font color="#0000FF">(3 players)</font>
<font color="#CC3333">Hero bets</font>, <font color="#CC3333">BB raises</font>, <font color="#CC3333">CO 3-bets</font>, Hero calls, BB calls.

Turn: (7.50 BB) 3[img]/images/graemlins/diamond.gif[/img] <font color="#0000FF">(3 players)</font>
Hero checks, BB checks, <font color="#CC3333">CO bets</font>, Hero calls, BB calls.

River: (10.50 BB) 9[img]/images/graemlins/diamond.gif[/img] <font color="#0000FF">(3 players)</font>
Hero checks, BB checks, <font color="#CC3333">CO bets</font>, Hero calls, BB folds.

Final Pot: 12.50 BB

[shadow29]

Cap the flop, then play.

[FlopMe]

I'm capping the flop, then leading the turn. If raised again, calling down.

[gonzopro]

[ QUOTE ]
I'm capping the flop, then leading the turn. If raised again, calling down.

[/ QUOTE ]

Exactly. I see OESD and flush draws 3 betting the flop here. Since neither hit on the turn, I keep firing.